I mainly watch animes that have a bit of humor to them, but this is an exception. Death Note is definitely more serious and I suppose could be considered a psychological thriller.
As with almost all anime, it follows a young high school boy. His name is Light and one day he stumbles upon a "Death Note" dropped by a shinigami (Japanese for "Death God) named Ryuk. Any name can be written in the Death Note and that person will die (for example, he could write that John Smith will die in 2 hours in a car accident. There are more rules to it but that's the main idea). Light uses this notebook to try to rid the world of evil by killing only criminals, but as you can imagine he gets a little carried away when he starts to think of himself as God.
Since the police kind of can't ignore that all the criminals in the city are mysteriously dieing, they hire the services of "L" who is a young super detective of sorts (who of course only eats cake and sweets). L's involvement becomes a problem for Light and a psychological battle insues.
This is actually a pretty deep anime. It touches on many moral issues and forces you to really think about what you would do in the same situation. In China, it was actually banned because kids were making their own "Death Notes" and writing the names of their enemies and teachers in it. You can pretty much chalk that up to a bunch of emo teenagers, but if you had a Death Note, what would you do with it? Would you use it for good, evil, or not use it at all?
If you're interested in watching, you can do so here on Hulu for free (along with a ton of other anime).